## A Companion to Analysis: A Second First and First Second by T. W. Korner

By T. W. Korner

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Extra resources for A Companion to Analysis: A Second First and First Second Course in Analysis

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We thus have B(x, δ) ⊆ f −1 (U ). It follows that f −1 (U ) is open. Sufficiency Suppose that f −1 (U ) is open whenever U is an open subset of Rp . Let x ∈ Rm and > 0. Since B(f (x), ) is open, it follows that f −1 (B(f (x), )) is open. But x ∈ f −1 (B(f (x), )), so there exists a δ > 0 such that B(x, δ) ⊆ f −1 (B(f (x), )). We thus have f (x) − f (y) < whenever x − y < δ. It follows that f is continuous. 18. Show that sin((−5π, 5π)) = [−1, 1]. Give examples of bounded open sets A in R such that (a) sin A is closed and not open, (b) sin A is open and not closed, (c) sin A is neither open nor closed, (d) sin A is open and closed.

Since we have shown that it is impossible that c = a or a < c < b, it follows that c = b. Since the supremum of a set need not belong to that set we must still prove that b ∈ E. However, if we choose a t0 ≥ a with b > t0 > b − δ the arguments of the previous paragraph give f (b) − f (t0 ) ≤ (K + /2)(b − t0 ) and f (t0 ) − f (a) ≤ (K + )(t0 − a), so f (c) ≤ (K + )(c − a). The arguments of the previous paragraph also give f (t) − f (a) ≤ (K + )(t − a) for a ≤ t < c, so c ∈ E and the theorem follows.

State and prove a similar result for lim inf. Although we mention lim sup from time to time, we shall not make great use of the concept. The reader will not be surprised to learn that the Bolzano-Weierstrass theorem is precisely equivalent to the fundamental axiom. 9. Suppose that F is an ordered field for which the BolzanoWeierstrass theorem holds (that is, every bounded sequence has a convergent subsequence). Suppose that an is an increasing sequence bounded above. Use the Bolzano-Weierstrass theorem to show that there exists an a ∈ F and n(1) < n(2) < .